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Theorem relopabVD 31634
Description: Virtual deduction proof of relopab 4964. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. relopab 4964 is relopabVD 31634 without virtual deductions and was automatically derived from relopabVD 31634.
1::  |-  (. y  =  v  ->.  y  =  v ).
2:1:  |-  (. y  =  v  ->.  <. x ,. y >.  =  <. x ,. v  >. ).
3::  |-  (. y  =  v ,. x  =  u  ->.  x  =  u ).
4:3:  |-  (. y  =  v ,. x  =  u  ->.  <. x ,. v >.  =  <.  u ,  v >. ).
5:2,4:  |-  (. y  =  v ,. x  =  u  ->.  <. x ,. y >.  =  <.  u ,  v >. ).
6:5:  |-  (. y  =  v ,. x  =  u  ->.  ( z  =  <. x ,. y  >.  ->  z  =  <. u ,  v >. ) ).
7:6:  |-  (. y  =  v  ->.  ( x  =  u  ->  ( z  =  <. x ,.  y >.  ->  z  =  <. u ,  v >. ) ) ).
8:7:  |-  ( y  =  v  ->  ( x  =  u  ->  ( z  =  <. x ,. y  >.  ->  z  =  <. u ,  v >. ) ) )
9:8:  |-  ( E. v y  =  v  ->  E. v ( x  =  u  ->  ( z  =  <. x ,  y >.  ->  z  =  <. u ,  v >. ) ) )
90::  |-  ( v  =  y  <->  y  =  v )
91:90:  |-  ( E. v v  =  y  <->  E. v y  =  v )
92::  |-  E. v v  =  y
10:91,92:  |-  E. v y  =  v
11:9,10:  |-  E. v ( x  =  u  ->  ( z  =  <. x ,. y >.  ->  z  =  <. u ,  v >. ) )
12:11:  |-  ( x  =  u  ->  E. v ( z  =  <. x ,. y >.  ->  z  =  <. u ,  v >. ) )
13::  |-  ( E. v ( z  =  <. x ,. y >.  ->  z  =  <. u  ,  v >. )  ->  ( z  =  <. x ,  y >.  ->  E. v z  =  <. u ,  v >. ) )
14:12,13:  |-  ( x  =  u  ->  ( z  =  <. x ,. y >.  ->  E. v  z  =  <. u ,  v >. ) )
15:14:  |-  ( E. u x  =  u  ->  E. u ( z  =  <. x ,. y  >.  ->  E. v z  =  <. u ,  v >. ) )
150::  |-  ( u  =  x  <->  x  =  u )
151:150:  |-  ( E. u u  =  x  <->  E. u x  =  u )
152::  |-  E. u u  =  x
16:151,152:  |-  E. u x  =  u
17:15,16:  |-  E. u ( z  =  <. x ,. y >.  ->  E. v z  =  <.  u ,  v >. )
18:17:  |-  ( z  =  <. x ,. y >.  ->  E. u E. v z  =  <.  u ,  v >. )
19:18:  |-  ( E. y z  =  <. x ,. y >.  ->  E. y E. u  E. v z  =  <. u ,  v >. )
20::  |-  ( E. y E. u E. v z  =  <. u ,. v >.  ->  E. u E. v z  =  <. u ,  v >. )
21:19,20:  |-  ( E. y z  =  <. x ,. y >.  ->  E. u E. v z  =  <. u ,  v >. )
22:21:  |-  ( E. x E. y z  =  <. x ,. y >.  ->  E. x  E. u E. v z  =  <. u ,  v >. )
23::  |-  ( E. x E. u E. v z  =  <. u ,. v >.  ->  E. u E. v z  =  <. u ,  v >. )
24:22,23:  |-  ( E. x E. y z  =  <. x ,. y >.  ->  E. u  E. v z  =  <. u ,  v >. )
25:24:  |-  { z  |  E. x E. y z  =  <. x ,. y >. }  C_  { z  |  E. u E. v z  =  <. u ,  v >. }
26::  |-  x  e.  _V
27::  |-  y  e.  _V
28:26,27:  |-  ( x  e.  _V  /\  y  e.  _V )
29:28:  |-  ( z  =  <. x ,. y >.  <->  ( z  =  <. x ,. y  >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) )
30:29:  |-  ( E. y z  =  <. x ,. y >.  <->  E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) )
31:30:  |-  ( E. x E. y z  =  <. x ,. y >.  <->  E. x  E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) )
32:31:  |-  { z  |  E. x E. y z  =  <. x ,. y >. }  =  {  z  |  E. x E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) }
320:25,32:  |-  { z  |  E. x E. y ( z  =  <. x ,. y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) }  C_  { z  |  E. u E. v z  =  <. u ,  v >. }
33::  |-  u  e.  _V
34::  |-  v  e.  _V
35:33,34:  |-  ( u  e.  _V  /\  v  e.  _V )
36:35:  |-  ( z  =  <. u ,. v >.  <->  ( z  =  <. u ,. v  >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) )
37:36:  |-  ( E. v z  =  <. u ,. v >.  <->  E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) )
38:37:  |-  ( E. u E. v z  =  <. u ,. v >.  <->  E. u  E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) )
39:38:  |-  { z  |  E. u E. v z  =  <. u ,. v >. }  =  {  z  |  E. u E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) }
40:320,39:  |-  { z  |  E. x E. y ( z  =  <. x ,. y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) }  C_  { z  |  E. u E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) }
41::  |-  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V  ) }  =  { z  |  E. x E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) )  }
42::  |-  { <. u ,. v >.  |  ( u  e.  _V  /\  v  e.  _V  ) }  =  { z  |  E. u E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) )  }
43:40,41,42:  |-  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V  ) }  C_  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }
44::  |-  { <. u ,. v >.  |  ( u  e.  _V  /\  v  e.  _V  ) }  =  ( _V  X.  _V )
45:43,44:  |-  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V  ) }  C_  ( _V  X.  _V )
46:28:  |-  ( ph  ->  ( x  e.  _V  /\  y  e.  _V ) )
47:46:  |-  { <. x ,. y >.  |  ph }  C_  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V ) }
48:45,47:  |-  { <. x ,. y >.  |  ph }  C_  ( _V  X.  _V )
qed:48:  |-  Rel  { <. x ,. y >.  |  ph }
(Contributed by Alan Sare, 9-Jul-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
relopabVD  |-  Rel  { <. x ,  y >.  |  ph }

Proof of Theorem relopabVD
Dummy variables  z 
v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2973 . . . . . 6  |-  x  e. 
_V
2 vex 2973 . . . . . 6  |-  y  e. 
_V
31, 2pm3.2i 455 . . . . 5  |-  ( x  e.  _V  /\  y  e.  _V )
43a1i 11 . . . 4  |-  ( ph  ->  ( x  e.  _V  /\  y  e.  _V )
)
54ssopab2i 4614 . . 3  |-  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }
63biantru 505 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>. 
<->  ( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
76exbii 1634 . . . . . . . . 9  |-  ( E. y  z  =  <. x ,  y >.  <->  E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
87exbii 1634 . . . . . . . 8  |-  ( E. x E. y  z  =  <. x ,  y
>. 
<->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
98abbii 2553 . . . . . . 7  |-  { z  |  E. x E. y  z  =  <. x ,  y >. }  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e. 
_V  /\  y  e.  _V ) ) }
10 ax6ev 1710 . . . . . . . . . . . . . . 15  |-  E. u  u  =  x
11 equcom 1732 . . . . . . . . . . . . . . . 16  |-  ( u  =  x  <->  x  =  u )
1211exbii 1634 . . . . . . . . . . . . . . 15  |-  ( E. u  u  =  x  <->  E. u  x  =  u )
1310, 12mpbi 208 . . . . . . . . . . . . . 14  |-  E. u  x  =  u
14 ax6ev 1710 . . . . . . . . . . . . . . . . . . 19  |-  E. v 
v  =  y
15 equcom 1732 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  y  <->  y  =  v )
1615exbii 1634 . . . . . . . . . . . . . . . . . . 19  |-  ( E. v  v  =  y  <->  E. v  y  =  v )
1714, 16mpbi 208 . . . . . . . . . . . . . . . . . 18  |-  E. v 
y  =  v
18 idn1 31284 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (. y  =  v  ->.  y  =  v ).
19 opeq2 4058 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  =  v  ->  <. x ,  y >.  =  <. x ,  v >. )
2018, 19e1a 31346 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (. y  =  v  ->.  <. x ,  y
>.  =  <. x ,  v >. ).
21 idn2 31332 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (. y  =  v ,. x  =  u  ->.  x  =  u ).
22 opeq1 4057 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  u  ->  <. x ,  v >.  =  <. u ,  v >. )
2321, 22e2 31350 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (. y  =  v ,. x  =  u  ->.  <. x ,  v
>.  =  <. u ,  v >. ).
24 eqeq1 2447 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( <.
x ,  y >.  =  <. x ,  v
>.  ->  ( <. x ,  y >.  =  <. u ,  v >.  <->  <. x ,  v >.  =  <. u ,  v >. )
)
2524biimprd 223 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
x ,  y >.  =  <. x ,  v
>.  ->  ( <. x ,  v >.  =  <. u ,  v >.  ->  <. x ,  y >.  =  <. u ,  v >. )
)
2620, 23, 25e12 31454 . . . . . . . . . . . . . . . . . . . . . 22  |-  (. y  =  v ,. x  =  u  ->.  <. x ,  y
>.  =  <. u ,  v >. ).
27 eqeq2 2450 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
x ,  y >.  =  <. u ,  v
>.  ->  ( z  = 
<. x ,  y >.  <->  z  =  <. u ,  v
>. ) )
2827biimpd 207 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
x ,  y >.  =  <. u ,  v
>.  ->  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )
)
2926, 28e2 31350 . . . . . . . . . . . . . . . . . . . . 21  |-  (. y  =  v ,. x  =  u  ->.  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. ) ).
3029in2 31324 . . . . . . . . . . . . . . . . . . . 20  |-  (. y  =  v  ->.  ( x  =  u  ->  ( z  =  <. x ,  y
>.  ->  z  =  <. u ,  v >. )
) ).
3130in1 31281 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  v  ->  (
x  =  u  -> 
( z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) ) )
3231eximi 1625 . . . . . . . . . . . . . . . . . 18  |-  ( E. v  y  =  v  ->  E. v ( x  =  u  ->  (
z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) ) )
3317, 32ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  E. v
( x  =  u  ->  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )
)
343319.37aiv 1919 . . . . . . . . . . . . . . . 16  |-  ( x  =  u  ->  E. v
( z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) )
35 19.37v 1918 . . . . . . . . . . . . . . . . 17  |-  ( E. v ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )  <->  ( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3635biimpi 194 . . . . . . . . . . . . . . . 16  |-  ( E. v ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )  ->  ( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3734, 36syl 16 . . . . . . . . . . . . . . 15  |-  ( x  =  u  ->  (
z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3837eximi 1625 . . . . . . . . . . . . . 14  |-  ( E. u  x  =  u  ->  E. u ( z  =  <. x ,  y
>.  ->  E. v  z  = 
<. u ,  v >.
) )
3913, 38ax-mp 5 . . . . . . . . . . . . 13  |-  E. u
( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
403919.37aiv 1919 . . . . . . . . . . . 12  |-  ( z  =  <. x ,  y
>.  ->  E. u E. v 
z  =  <. u ,  v >. )
4140eximi 1625 . . . . . . . . . . 11  |-  ( E. y  z  =  <. x ,  y >.  ->  E. y E. u E. v  z  =  <. u ,  v
>. )
42 19.9v 1717 . . . . . . . . . . . 12  |-  ( E. y E. u E. v  z  =  <. u ,  v >.  <->  E. u E. v  z  =  <. u ,  v >.
)
4342biimpi 194 . . . . . . . . . . 11  |-  ( E. y E. u E. v  z  =  <. u ,  v >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4441, 43syl 16 . . . . . . . . . 10  |-  ( E. y  z  =  <. x ,  y >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4544eximi 1625 . . . . . . . . 9  |-  ( E. x E. y  z  =  <. x ,  y
>.  ->  E. x E. u E. v  z  =  <. u ,  v >.
)
46 19.9v 1717 . . . . . . . . . 10  |-  ( E. x E. u E. v  z  =  <. u ,  v >.  <->  E. u E. v  z  =  <. u ,  v >.
)
4746biimpi 194 . . . . . . . . 9  |-  ( E. x E. u E. v  z  =  <. u ,  v >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4845, 47syl 16 . . . . . . . 8  |-  ( E. x E. y  z  =  <. x ,  y
>.  ->  E. u E. v 
z  =  <. u ,  v >. )
4948ss2abi 3422 . . . . . . 7  |-  { z  |  E. x E. y  z  =  <. x ,  y >. }  C_  { z  |  E. u E. v  z  =  <. u ,  v >. }
509, 49eqsstr3i 3385 . . . . . 6  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V )
) }  C_  { z  |  E. u E. v  z  =  <. u ,  v >. }
51 vex 2973 . . . . . . . . . . 11  |-  u  e. 
_V
52 vex 2973 . . . . . . . . . . 11  |-  v  e. 
_V
5351, 52pm3.2i 455 . . . . . . . . . 10  |-  ( u  e.  _V  /\  v  e.  _V )
5453biantru 505 . . . . . . . . 9  |-  ( z  =  <. u ,  v
>. 
<->  ( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5554exbii 1634 . . . . . . . 8  |-  ( E. v  z  =  <. u ,  v >.  <->  E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5655exbii 1634 . . . . . . 7  |-  ( E. u E. v  z  =  <. u ,  v
>. 
<->  E. u E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5756abbii 2553 . . . . . 6  |-  { z  |  E. u E. v  z  =  <. u ,  v >. }  =  { z  |  E. u E. v ( z  =  <. u ,  v
>.  /\  ( u  e. 
_V  /\  v  e.  _V ) ) }
5850, 57sseqtri 3386 . . . . 5  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V )
) }  C_  { z  |  E. u E. v ( z  = 
<. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V )
) }
59 df-opab 4349 . . . . 5  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) }
60 df-opab 4349 . . . . 5  |-  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }  =  { z  |  E. u E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) }
6158, 59, 603sstr4i 3393 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  C_  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }
62 df-xp 4844 . . . . 5  |-  ( _V 
X.  _V )  =  { <. u ,  v >.  |  ( u  e. 
_V  /\  v  e.  _V ) }
6362eqcomi 2445 . . . 4  |-  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }  =  ( _V  X.  _V )
6461, 63sseqtri 3386 . . 3  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  C_  ( _V  X.  _V )
655, 64sstri 3363 . 2  |-  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V )
66 df-rel 4845 . . 3  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  <->  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V ) )
6766biimpri 206 . 2  |-  ( {
<. x ,  y >.  |  ph }  C_  ( _V  X.  _V )  ->  Rel  { <. x ,  y
>.  |  ph } )
6865, 67e0a 31502 1  |-  Rel  { <. x ,  y >.  |  ph }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2427   _Vcvv 2970    C_ wss 3326   <.cop 3881   {copab 4347    X. cxp 4836   Rel wrel 4843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-opab 4349  df-xp 4844  df-rel 4845  df-vd1 31280  df-vd2 31288
This theorem is referenced by: (None)
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